Abstract

Optical Fourier processing is carried out by placing volume-hologram filters in the object plane of an optical system instead of in the conventional Fourier plane. This new type of Fourier processing is analyzed, and experimental results of optical differentiation by a volume-hologram filter are shown.

© 1979 Optical Society of America

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References

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  1. D. Peri, A. A. Friesem, Opt. Lett. 3, 124 (1978).
    [CrossRef] [PubMed]
  2. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  3. S. K. Case, Multiple Exposure Holography in Volume Materials, thesis, University of Michigan, 1976 (University Microfilms, Inc., order # 76-27,461).
  4. P. Hutzler, Appl Opt. 16, 2264 (1977).
    [CrossRef] [PubMed]
  5. The odd symmetry of Eq. (7) about ν = 0 merely indicates that the volume grating distinguishes between deviations from the Bragg angle by positive or negative angles.
  6. J. K. T. Eu, C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973); Errata, Opt. Commun. 10, 211 (1974).
    [CrossRef]
  7. S. K. Case, W. J. Dallas, Appl. Opt. 17, 2537 (1978).
    [PubMed]

1978 (2)

1977 (1)

P. Hutzler, Appl Opt. 16, 2264 (1977).
[CrossRef] [PubMed]

1973 (1)

J. K. T. Eu, C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973); Errata, Opt. Commun. 10, 211 (1974).
[CrossRef]

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Case, S. K.

S. K. Case, W. J. Dallas, Appl. Opt. 17, 2537 (1978).
[PubMed]

S. K. Case, Multiple Exposure Holography in Volume Materials, thesis, University of Michigan, 1976 (University Microfilms, Inc., order # 76-27,461).

Dallas, W. J.

Eu, J. K. T.

J. K. T. Eu, C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973); Errata, Opt. Commun. 10, 211 (1974).
[CrossRef]

Friesem, A. A.

Hutzler, P.

P. Hutzler, Appl Opt. 16, 2264 (1977).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Liu, C. Y. C.

J. K. T. Eu, C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973); Errata, Opt. Commun. 10, 211 (1974).
[CrossRef]

Lohmann, A. W.

J. K. T. Eu, C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973); Errata, Opt. Commun. 10, 211 (1974).
[CrossRef]

Peri, D.

Appl Opt. (1)

P. Hutzler, Appl Opt. 16, 2264 (1977).
[CrossRef] [PubMed]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Opt. Commun. (1)

J. K. T. Eu, C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973); Errata, Opt. Commun. 10, 211 (1974).
[CrossRef]

Opt. Lett. (1)

Other (2)

S. K. Case, Multiple Exposure Holography in Volume Materials, thesis, University of Michigan, 1976 (University Microfilms, Inc., order # 76-27,461).

The odd symmetry of Eq. (7) about ν = 0 merely indicates that the volume grating distinguishes between deviations from the Bragg angle by positive or negative angles.

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Figures (4)

Fig. 1
Fig. 1

Optical system for Fourier processing in the object plane.

Fig. 2
Fig. 2

Filter-intensity transmittance versus spatial frequency.

Fig. 3
Fig. 3

Input object.

Fig. 4
Fig. 4

Differentiated output.

Equations (8)

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θ = λ ν ,
I ( x , y ) = F [ ν ( x , y ) , μ ( x , y ) ] O ˜ ( ν , μ ) exp [ - 2 π i ( ν x + μ y ) ] d ν d μ ,
F = F T ( ν , μ ) + F D ( ν , μ ) exp [ - 2 π i ( ν 0 x cos γ + ν 0 y sin γ ) ] .
F D ( ν ) = - i α sin [ α 2 + β 2 ( ν - ν B ) 2 ] 1 / 2 [ α 2 + β 2 ( ν - ν B ) 2 ] 1 / 2 ,
F T ( ν ) = cos [ α 2 + β 2 ( ν - ν B ) 2 ] 1 / 2 + i β ( ν - ν B ) sin [ α 2 + β 2 ( ν - ν B ) 2 ] 1 / 2 [ α 2 + β 2 ( ν - ν B ) 2 ] 1 / 2 .
F T ( ν ) = - β 2 ν 2 π + i 2 π β ν .
F T ( ν ) i 2 π β ν = ( β π 2 ) 2 π i ν .
I ( x , y ) = ( β π 2 ) x 0 ( x , y ) ,

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